LdiJ, if your question is did the landlord fail to mitigate as of June 1, I believe that the answer is yes and is, therefore, not entitled to any rent from June 1 onward.
The Iowa Supreme Court made that clear in Vawter v. McKissick (1968).
Defendant/Tenant McKissick abandoned the premises on December 31 1965, delivering the keys to Plaintiff/Landlord Vawter that day. Landlord waited until the lease ran out on May 31, 1966 and then sue for all the rent. The trial court ruled in favor of the tenants.
Some quotes from the case:
"In this state we are committed to the doctrine that when a tenant wrongfully abandons leased premises, the landlord is under a duty to show reasonable diligence has been used to relet the property at the best obtainable rent and thereby obviate or reduce the resulting damage"
"The trial court held plaintiff had an affirmative obligation to show she did something beginning January 1, 1966, to try to relet the leased premises."
"The burden of proof was on the appellees [landlord] to show diligence in reletting the property."
"The court stated it was his impression from hearing plaintiff's testimony she was more than happy to let the lease run out and attempt to collect the balance of the rent from defendants; and it was significant that while she claimed she was unable to rent the building during the rest of the term, she was able to rent it within seven days after the lease expired.
The action of the trial court in dismissing plaintiff's petition and giving her the abandoned property covered by the attachment is
Affirmed."
https://scholar.google.com/scholar_case?case=17494708805208206716&q=landlord+mitigation&hl=en&as_sdt=4,16
The landlord's duty to mitigate was later codified in Iowa statute 562A.3:
https://law.justia.com/codes/iowa/2018/title-xiv/chapter-562a/section-562a.29/
In DR Mobile Home Rentals v. Frost (1996) the court wrote:
"We can find no reason to believe that the enactment of chapter 562A in any way changed the common-law rule that "a showing of diligence in reletting the premises is an essential element of [a landlord's] right to recover."
Vawter v. McKissick, 159 N.W.2d 538, 542 (Iowa 1968).
[1] We therefore reaffirm our
Vawter holding: "
n an action brought by a landlord against a tenant who has wrongfully abandoned the leased premises prior to the expiration of the term, the landlord must allege and prove what [was done] in attempting to rerent the premises after [the landlord] knew or should have known they were abandoned.""
"We reverse the award for unpaid rent because the landlord failed to allege and prove what it did in attempting to rerent the abandoned unit."
https://scholar.google.com/scholar_case?case=1473768050980937534&hl=en&as_sdt=1000005&sciodt=4,16
In other states, both statute and case law indicate that converting the rental to the landlord's own use, i.e.: selling it without re-renting constitutes failure to mitigate.
I didn't go any further than the two cases that cited but my guess is that Iowa courts might rule similarly.
The amount at issue for OP is likely within the Iowa small claims limit of $6500.
(I don't know why this changed to italics. The gremlins won't let me put it back.)